(i) Boosting with Decision Trees. 
The DT boosted densities with the exact fairness leveraging scheme \( \vartheta ^{E}_{t} \)  maintain a high RR, with \(\textsc {RR}(Q^{\mathrm {DT}}_{T}) > 0.94 \)  for both real-world datasets with either fairness parameter \( \tau = 0.7, 0.9\) . Furthermore, the difference between KL for the different fairness parameters are minimal between these two datasets, with test KL differing by only 0.005 for COMPAS with the sex sensitive attribute. It can also be seen in Figure REF  that the RR does not change much after 5 boosting iterations on this dataset. On the other hand, in the synthetic dataset the final RR of \( Q^{\mathrm {DT}}_{T} \)  are very close to the \( \tau \)  parameter of FBDE.
The DT boosted densities with relative fairness leveraging scheme \( \vartheta ^{R}_{t} \)  follow a similar story for the real-world datasets. Despite having a significantly weaker fairness guarantee in Theorem REF , \(\textsc {RR}(Q^{\mathrm {DT}}_{T}) > 0.92 \)  in these datasets. The KL is also consistently lower than the corresponding exact fairness boosting density. The weaker guarantee becomes apparent in the synthetic results, where the final RR becomes lower than the parameter \( \tau \)  of FBDE. As suggested by Figure REF , further decay of the RR occurs after additional boosting iterations, explicitly shown in the Appendix.
When comparing to the baseline approaches from [1]}, we often have lower fairness metrics but better KL accuracy. In particular, the RR is often slightly lower than the DT approaches – particularly when using the relative leveraging scheme. The RR is much higher across the board than our approaches, given that the max-ent method aims to ensure that this is high. However, this does not come without a cost. The KL is much higher for our approach, with the largest difference appearing in the Adult dataset; where we achieve a test KL of \( ~0.088 \)  versus \( 0.483 \)  and \( 0.901 \)  for \( \texttt {M.Ent} \)  \( \theta ^{c} \)  and \( \texttt {M.Ent} \)  \( \theta ^{b} \) , respectively.
